Ultra-high-energy gamma rays are found to come from the Equator of our galaxy, with energies up to \(3.5 \cdot 10^{12} \mathrm{eV}\). What is the wavelength of this light? How does the energy of this light compare to the rest mass of a proton?

Given energy, \(E = 3.5 \cdot 10^{12} \mathrm{eV}\). To convert it into SI units (joules), we need to multiply it by the charge of an electron. \(1 \mathrm{eV} = 1.6 \cdot 10^{-19} \mathrm{J}\) So, \(E = 3.5 \cdot 10^{12} \mathrm{eV} \times 1.6 \cdot 10^{-19} \frac{\mathrm{J}}{\mathrm{eV}} = 5.6 \cdot 10^{-7} \mathrm{J}\)

We will use the energy-wavelength relationship given by Planck's constant (\(h\)) and the speed of light (\(c\)). For a photon, the energy-wavelength relationship is given by \(E = h \cdot c / \lambda\) where \(E\) is the energy, and \(\lambda\) is the wavelength. Rearranging for \(\lambda\), we get \(\lambda = h \cdot c / E\) Using the values for Planck's constant (\(h = 6.63 \cdot 10^{-34} \mathrm{Js}\)) and the speed of light (\(c = 3.0 \cdot 10^{8} \mathrm{m/s}\)), we can compute the wavelength as follows: \(\lambda = \frac{6.63 \cdot 10^{-34} \mathrm{Js} \times 3.0 \cdot 10^{8} \mathrm{m/s}}{5.6 \cdot 10^{-7} \mathrm{J}} = 3.56 \cdot 10^{-17} \mathrm{m}\)

To compare the energy of this light with the rest mass of a proton, we need to convert the proton mass into energy using the mass-energy equivalence formula. Einstein's mass-energy equivalence formula is given by \(E = m \cdot c^2\) where \(m\) is the rest mass of a proton, which is approximately \(1.67 \cdot 10^{-27} \mathrm{kg}\), and \(c\) is the speed of light, \(3.0 \cdot 10^{8} \mathrm{m/s}\). The energy of the proton is given by \(E_\mathrm{proton} = 1.67 \cdot 10^{-27} \mathrm{kg} \times (3.0 \cdot 10^{8} \mathrm{m/s})^2 = 1.50 \cdot 10^{-10} \mathrm{J}\) Now, we will compare this energy with the energy of the gamma ray by taking their ratio \(\frac{E_{\mathrm{gamma}}}{E_\mathrm{proton}} = \frac{5.6 \cdot 10^{-7} \mathrm{J}}{1.50 \cdot 10^{-10} \mathrm{J}} = 3733\) The energy of the ultra-high-energy gamma ray is 3733 times the rest mass energy of a proton.